Hess: (Joint work with J. Rognes) Let t be a twisting cochain from a connected, coaugmented chain coalgebra C to an augmented chain algebra A over an arbitrary PID R. I'll explain the construction of a twisted extension of chain complexes H(t) of which both the Hochschild complex of an associative algebra and the coHochschild complex of a coassociative coalgebra are special cases. We call H(t) the Hochschild complex of t.

When A is a chain Hopf algebra, I'll give conditions under which H(t) admits an rth-power map extending the usual rth-power map on A and lifting the identity on C. In particular, both the Hochschild complex of any cocommutative Hopf algebra and the coHochschild complex of the normalized chain complex of a double suspension admit power maps. Moreover, if K is a double
suspension, then the power map on the coHochschild complex of the normalized chain complex of K is a model for the topological power map on the free loops on K, illustrating the topological relevance of our algebraic construction.

The first and second lectures will be devoted to recalling the necessary algebraic background material, as well as the history of the rth-power maps in algebra and topology, then to explaining the construction and algebraic properties of the Hochschild construction H(t). In the third lecture I will explain the link with topology.

Dwyer: Lecture I will be an introduction to homotopy theories and model categories, as well as to homotopy limits and colimits. During Lecture II, I will talk about how to get spaces from categories and about localization. Finally, the subjects of Lecture III will be cohomology of function spaces and maps between classifying spaces.

Dror-Farjoun : (Joint work with Dwyer and Prezma.) The localization and cellularization of a principal fibration G--->X-->B is still a principal fibration under mild conditions, although the nature of the new group that arises is still hard to unlock. One way to approach this problem is by means of "normal maps" of loop spaces, which is a homotopy version of a normal subgroup. We show how to build a Segal-like machine that characterizes normal maps and that enables us to show they are preserved under product-preserving homotopy functors.

Skoda : Nonabelian cocycles are related to weak functors between
higher categories; thus their combinatorics is difficult when the
degree is large, say if already larger than 2. However, in most
applications the higher categories themselves can be taken to be strict,
which is a great simplification, while the weak functors are still an issue.

The category of strict omega-categories has a Quillen
model category structure which has been recently studied. Higher nonabelian
cocycles of a space X with coefficients in a presheaf of omega-categories A
by our definition form an omega-category which is enriched hom
from a cofibrant replacement of some presheaf of omega categories
related to X (or a (hyper)cover of X; e.g. fundamental, path and
Cech groupoids are examples) to A.
Sometimes, one needs to use additional colimit procedures
(e.g. for refinement of covers). The cofibrant replacement involves
combinatorics similar to Street orientals; however the latter suffice only
if the coefficient presheaf A is a presheaf of omega-groupoids.
This is a work in progress with U. Schreiber, D. Stevenson and H. Sati.

Seal : After reviewing the theory of monads, we will investigate how certain
monads on SET can be modified to become monads on categories of
Kleisli monoids (these include for example the category ORD of ordered
sets, or TOP of topological spaces). The new monads turn out to have
the same Eilenberg-Moore category as the original ones, but allow for
a more transparent description of the corresponding Eilenberg-Moore
algebras. This simplified presentation will lead us to the
identification of injective objects in the base category of Kleisli
monoids.

Shamir : Local cohomology has proved to be a useful tool in commutative
algebra. Local cohomology has also made surprising appearances in
algebraic topology, as in the works of Benson and Greenlees and
others. For example, Benson and Greenlees' results provide a sort of
duality statement for the cohomology of certain classifying spaces.
I will "define" local cohomology in simple terms (in fact I will use
Dwyer and Greenlees' equivalent definition) and explain some of its
uses in algebraic topology. I will also offer a way to generalize some
of these results into a non-commutative setting.

van der Zypen : Priestley duality is a "dictionary" between distributive
lattices and compact, totally order-disconnected topological spaces
(Priestley spaces). We give an introduction as well as examples for
lattice theoretical problems that are more treatable when "translated"
into the language of Priestley spaces. Finally, we briefly discuss
some open problems.

Ching : I'll explain joint work with Greg Arone that decomposes the Goodwillie
tower of a functor from spaces to spaces. We construct an
approximation to the tower built from a bimodule structure on the
derivatives of the functor, and show that the fibre of the map from
the real tower to the approximation can be described in terms of Tate
spectra. In particular, in cases where the Tate spectra vanish, such
as rationally, we obtain models for the Goodwillie tower explicitly in
terms of this bimodule. I'll also mention our plans to apply a result
of Nick Kuhn on vanishing Tate spectra to this setting.

Genton : In order to study the classifying space of a group G, a filtration by principal G-bundles of the universal G-bundle is constructed using the spaces of homomorphisms from the descending central series into G and a simplicial structure similar to the bar construction of classifying spaces. The second stage of this filtration is built upon the commuting elements Z^n -> G. An isomorphism allows us to compute its rational cohomology using the maximal torus of Lie group G. SU(2) is given as an example.

Lack : There is a well-known Quillen model structure on the category Cat
of categories and functors for which the weak equivalences are the
equivalences of categories. This restricts to a model structure on
the full subcategory Gpd of Cat consisting of the groupoids, and this
provides a model for (not necessarily connected) homotopy 1-types.
I will describe how this "n=1" case (1-categories, 1-groupoids, 1-types)
can be extended to the cases of n=2 and n=3. In each of these cases there
is a model structure on the category of all n-dimensional categories. Once
again, there is a restricted model structure on the full subcategory
consisting of those n-dimensional categories for which all arrows at all
dimensions are invertible; and this provides a model for n-types. In the
case n=3, the n-dimensional categories considered are the Gray-categories
of the title.
I will build up from n=1 to n=2 and n=3 in an inductive sort of way,
although I do not know how to continue the induction to deal with higher n.

Lesh : I will discuss connections between the calculus of
functors and the Whitehead Conjecture, both for the classical
theorem of Kuhn and Priddy for symmetric powers of spheres and for
the analogous conjecture in topological K-theory. It turns out that
key constructions in Kuhn and Priddy's proof have bu-analogues, and
there is a surprising connection to the stable rank filtration of
algebraic K-theory.